which are suitably compatible with one another. The full set of coherence conditions may be summarized by saying TT preserves the two-sided monoidal action of VV on itself, in an appropriate 2-categorical sense. More precisely: the two-sided action of VV on itself is a lax functor of 2-categories

V˜:BV×(BV)op→Cat\tilde{V} \colon B V \times (B V)^{op} \to Cat

(BVB V is the one-object 2-category associated with a monoidal category VV, and (BV)op(B V)^{op} is the same 2-category but with 1-cell composition (= tensoring) in reverse order), and the two-sided strength means we have a structure of lax natural transformationV˜→V˜\tilde{V} \to \tilde{V}.

Remark

In the setting where VV is symmetric monoidal, we will assume that the left and right strengths τ\tau and σ\sigma are related by the symmetry in the obvious way, by a commutative square

There is a category of strong functors V→VV \to V, where the morphisms are transformations λ:S→T\lambda \colon S \to T which are compatible with the strengths in the obvious sense. Under composition, this is a strict monoidal category.

are both equal to αA,B\alpha_{A, B}. We show this for the first composite; the proof is similar for the second. If αT\alpha_T denotes the monoidal constraint for TT and αTT\alpha_{T T} the constraint for the composite TTT T, then by definition αTT\alpha_{T T} is the composite given by